 Hello and welcome to video number 17 of the online version of the future research lecture. You might remember that in the last two videos we talked about current driven instabilities. And in this video, we will start to talk about pressure driven instabilities. So pressure driven instabilities are topic of today's video instabilities. And the pressure driven instabilities are based on the interchange mechanism. So those are based on the interchange mechanism. Okay, so the interchange mechanism is what I've tried to, or what we will explain using the picture I have included here. So first of all, you see in the center, this is a polar cross section on the left on the high field side, where we have the pressure gradient pointing into the right direction. So the pressure gradient, this is the direction of the pressure gradient. And then the curvature radius also points into the right direction, whereas on the left hand side, which is the low field side of the torus of the polar cross section, we have the pressure gradient pointing to the left. And then the curvature radius pointing, of course, into the same direction. But this is now, well, in the same direction as on the high field side, but now pressure gradient and curvature radius are pointing into opposite directions. Okay. And you might remember from plasma physics one lecture, the curvature drift. So there's the curvature drift. The curvature drift basically was proportional to the curvature radius times the magnetic field direction divided by the charge curvature radius squared and magnetic field squared. Okay. Now let's first look at the left hand side. So first of all, the magnetic field points into the board here. So let's probably use something which is a bit more blue. Yeah. So this is a bit better. So this is the direction of the magnetic field pointing into the board. Good. Now with the magnetic field pointing into the board, the curvature radius pointing into the right hand side, the curvature drift for the ions is pointing upwards. So this is basically this corresponds to the direction of the ions of the curvature drift. Now the slightly darker color here should indicate regions of higher pressure, thus of higher plasma density than as opposed to the slightly lighter gray colored area. Remember the pressure gradient pointing to the right direction. Now this means that if we have an initial perturbation of something like a wave depicted here just by some initial random fluctuation, then due to the curvature drift, we have here more ions streaming upwards than in the region of the lighter plasma. This is why we have an enhancement of ions here, where we have a refraction of ions here leading to negative charge. And again, here we have an enhancement of ions and this now leads, of course, to electric fields, to electric fields, which are pointing here into this direction from plus to minus electric, sorry, electric field. And then electric field here pointing upwards, electric field. And as you well know, electric fields in a plasma with a background magnetic field. Here we have E cross B, this leads to an E cross B drift and the top pointing to the right. So here we have an E cross B drift pointing to the right. So this suppresses, this stabilizes the initial perturbation, whereas on the bottom with electric field pointing into the other direction, oops, sorry, with electric field pointing into the other direction, the E cross B drift also points into the other direction, again stabilizing the initial perturbation which we have here. So on the high field side, the initial perturbation is stabilized. Now let's have a look on the low field side. Once again, the magnetic field here, since we are in the same poloidal cross section, points into the board. So this is a magnetic field direction. Now this leads to the same direction for the curvature drift considering ions. So also going upwards. Now again, darker colored area corresponds to higher density, lighter colored area to less density, or less plasma pressure, thus also less density. We have an initial perturbation here, some kind of sinusoidal looking wave. And due to the fact that we have more ions on the left-hand side where the area is darker, then the drift leads to the fact that we have more ions here located in this area. And here we have less ions because more ions are moving upwards than coming from the bottom. Here again, we have less ions here because more ions have been moved away upwards than coming from the bottom here. This leads now again to electric fields pointing into this direction. So electric field pointing to this direction, here electric field pointing downwards into this direction, electric field pointing into this direction, electric fields in the plasma lead to E cross B drift. Now we have an E cross B drift, which goes on the top into the left direction. So going into this direction. So this is the direction of the E cross B drift. So meaning it enhances the initial perturbation. And the same is true on the bottom where the electric field then goes into the right direction, which enhances the initial perturbation. So here the initial perturbation is growing. And this is why on the right-hand side we call this also bad curvature. So this refers to the bad curvature region on the left-hand side. This is the good curvature region. So this is good curvature region. This is the interchange mechanism. Good. Now let's summarize what we have just learned. We have learned that plasma stability depends on relative directions of the pressure gradient and the curvature radius. So of pressure gradient and the magnetic field curvature basically, there exist regions of good curvature, there exist regions of good curvature and bad curvature and bad curvature. This is an expression you might find often literature, good curvature and bad curvature and refers essentially to the in-bord side of a toroidal experiment. So this is in-bord side and bad curvature is usually the out-bord side. Now instabilities driven by bad curvature correspond basically to the Rayleigh-Taylor fluid instability, which you might know from your hydrodynamics lecture. So instabilities driven by the bad curvature correspond to the Rayleigh-Taylor instability, to the Rayleigh-Taylor fluid instability I should say, which you're probably very familiar with. This instability now is therefore not surprisingly called the interchange instability. It's the interchange instability. Good. Now if we have a in-bord experiment modes which are elongated along field lines, of course they pass regions of good and bad curvature. This is why we need a flux surface average to properly treat it. So if we have some modes which are elongated along field lines, then this mode or these modes pass regions of good and bad curvature, which means for a proper analysis of the energy principle, to analyze if the mode is stable or unstable, we need to perform a flux surface average. We need to perform a flux surface average and you might remember shear, we talked already about shear and in these scenarios shear is good for stability, something worth to repeat here. I said that already in the beginning, in the beginning of chapter two when we talked about magnetic field configurations, but here as a reminder shear is good for stability because basically shear can de-correlate if you have some kind of instabilities here. Now let's first have a look at this group hinge. So before we go to the toroidal experiments, let's first have a look at this group hinge which is basically the same as saying we consider a cylindrical plasma. In such a plasma the pressure-driven modes basically have a constant amplitude along the magnetic field. There are no regions of good or bad curvature, so the amplitude stays the same. So the pressure-driven modes are basically constant, the amplitude is basically constant along the magnetic field lines since we have no regions of good or bad curvature. There's no good or bad curvature and in general of course in such a scenario, if we would have shear then shear would lead to stabilization, so stabilized, so these modes are stabilized by shear due to the same mechanism we explained when we talked first about shear. And when you apply the energy principle you basically find that all mn modes are stable if 4 of the following is fulfilled, 8 mu0 and then the radial pressure gradient and then the denominator we have rb phi squared. If this is smaller then 1 over qs and then dqs over dr, so basically the shear, so this is the stability criterion for a screw pinch and this is called the, oops, this is called the sui-darm criterion, the sui-darm criterion. And on the left hand side we basically have the destabilizing pressure drive, so here this one is basically the destabilizing pressure drive and on the right hand side here we have basically the shear stabilization, so this is the shear stabilization, this is the change of the safety factor with the radius which corresponds to the shear. Okay this is the case for a screw pinch, now what happens if you go to a torus, what happens if you go to a torus, then the criterion which we find there is called the mercier criterion, the mercier criterion and it reads so it is very similar 8 mu0, the radial pressure gradient then over rb phi squared and then the additional factor 1 minus qs squared, that this has to be smaller for stability, smaller than and then it's the same factor 1 qs times the radial derivative of the safety factor squared, so this is basically the same except for the factor 1 minus qs on the left hand side and in general one can say of course this is stable if qs is larger than 1 because then this inequality is always fulfilled and this is the same as saying or corresponds to saying that the average curvature should be or has to be sufficiently, sufficiently large such that the resultant, the resultant curvature is good such that the resultant curvature is good in the sense of good and bad curvature what we just explained a few slides ago and this is as you might know already no serious limitation in a tokamak this is no serious limitation let me whoops something wrong let me tation in tokamaks because qs is only smaller than one in the very center of tokamaks only in very center of tokamaks otherwise q is always the safety factor qs is always larger than one okay that's it for the pressure driven instabilities based on the interchange mechanism talking only about the screw pinch and sorry yeah the screw pinch from which the result from which the swedan criterion followed and also by bending the screw pitch into a torus the messier criterion followed from it and in the next video we will have a look what happens when we allow for a poloidal dependence of the amplitude of the initial perturbation then we will see so-called ballooning modes might arise imposing a limit on the plasma beta okay that's it hope to see you in the next video